Abstract
We compare the pencil/paper proofs and formal proof of two traditional proofs in high school geometry. We highlight the fact that slightly different formulations or proofs can lead to difficulties in the formalization. We discuss the challenges and impact on both mathematical teaching and on the design of AI tools for mathematical education.
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Notes
- 1.
A proof assistant is a piece of software which allows the user of the system to state mathematical definition and properties and to prove theorems interactively using a formal language. The proofs are checked mechanically.
- 2.
The comment in French Wikipedia about Amiot’s proof seems to say that the proof is valid only in Euclidean geometry because it uses the construction of parallel line AC trough B. To be precise, the proof does not rely on the uniqueness of this line only on its existence, so this first step of the proof is valid also in hyperbolic geometry (but not in elliptic geometry). The Wikipedia comment fails to notice that essential use of a version of the parallel postulate relies in the use of what we called above the postulate of alternate-interior angles.
- 3.
- 4.
We may add a definition of alternate-interior angles, which would be a shortcut for the predicate TS which states that two points are on opposite sides of a line, but adding more definitions make the formal proofs more cumbersome, that is why we hesitate to introduce a new definition.
- 5.
Note that the reciprocal is valid in neutral geometry.
- 6.
We have a separate lemma for this case, we could also assume that we have a proper triangle. In formal development, we always try to prove the most generic results.
- 7.
Note that we do need “the parallel line”, uniqueness is not important here.
- 8.
We could also use any point \(B_2\) such that B belongs to segment \(B_1B_2\).
- 9.
Maybe there is a simpler proof ? but for sure we need to use the fact that AC is parallel to \(B_1B_2\).
- 10.
Construction tools correspond to existence theorems.
- 11.
The proof could also be modified to construct \(B_1\) and \(B_2\) such that B belongs to segment \(B_1B_2\) and then say that at least one of them is on the opposite side of A with regard to the line BC.
- 12.
Excerpts from students’ productions are translated from French.
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Appendix: Verbatim of the Formal Proofs
Appendix: Verbatim of the Formal Proofs
We first give the formal proof of the fact that the sum of angles of a triangle is congruent to the flat angle.
Proof of Varignon’s theorem using the midpoint theorem:
Alternative proof using characterization of parallelogram using midpoints and coordinates:
A completely automatic proof using the area method:
A detailed proof script using the area method, the tactics highlights the key idea of eliminating points one by one from the goal, but the actual computation is implicit:
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Narboux, J., Durand-Guerrier, V. (2022). Combining Pencil/Paper Proofs and Formal Proofs, A Challenge for Artificial Intelligence and Mathematics Education. In: Richard, P.R., Vélez, M.P., Van Vaerenbergh, S. (eds) Mathematics Education in the Age of Artificial Intelligence. Mathematics Education in the Digital Era, vol 17. Springer, Cham. https://doi.org/10.1007/978-3-030-86909-0_8
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